Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

Let (X, d(X)) and (Y, d(Y)) be pointed compact metric spaces with distinguished base points e(X) and e(Y). The Banach algebra of all K-valued Lipschitz functions on X - where K is either C or R - that map the base point e(X) to 0 is denoted by Lip(0)(X). The peripheral range of a function f is an element of Lip(0)(X) is the set Ran(pi)(f) = {f(x) : vertical bar f(x)vertical bar = parallel to f parallel to(infinity)} of range values of maximum modulus. We prove that if T-1, T-2 : Lip(0)(X) -> Lip(0)(Y) and S-1, S-2 : Lip(0)(X) -> Lip(0)(X) are surjective mappings such that Ran(pi)(T-1(f) T-2(g)) boolean AND Ran(pi)(S-1(f) S-2(g)) not equal empty set for all f, g is an element of Lip(0)(X), then there are mappings phi(1), phi(2) : Y -> K with phi(1)(y)phi(2)(y) = 1 for all y is an element of Y and a base point-preserving Lipschitz homeomorphism psi : Y -> X such that T-j(f)(y) = phi(j)(y) S-j(f)(psi(y)) for all f is an element of Lip(0)(X), y is an element of Y, and j = 1, 2. In particular, if S-1 and S-2 are identity functions, then T-1 and T-2 are weighted composition operators.